论文标题
有限组的近似属性的不良位置
Bad places for the approximation property for finite groups
论文作者
论文摘要
给定一个数字字段$ k $和有限的$ k $ -group $ g $，$ g $的驯服近似问题询问限制映射$ h^1（k，g）\ to \ prod_ {v \inςh^1} $ \ text {bad} _g $是将$ g $的订单分配或在最小扩展中划分的订单的有限位置。在本文中，我们证明了集合$ \ text {bad} _g $是“敏锐的”。为此，我们证明有有限的Abelian $ k $ -groups $ a $其中地图$ h^1（k，a）\ to \ prod_ {将$ a $的订单分开，然后在最小扩展中分配$ a $。
Given a number field $k$ and a finite $k$-group $G$, the Tame Approximation Problem for $G$ asks whether the restriction map $H^1(k,G)\to\prod_{v\inΣ}H^1(k_v,G)$ is surjective for every finite set of places $Σ\subseteqΩ_k$ disjoint from $\text{Bad}_G$, where $\text{Bad}_G$ is the finite set of places that either divides the order of $G$ or ramifies in the minimal extension splitting $G$. In this paper we prove that the set $\text{Bad}_G$ is "sharp". To achieve this we prove that there are finite abelian $k$-groups $A$ where the map $H^1(k,A)\to\prod_{v\inΣ_0}H^1(k_v,A)$ is not surjective in a set $Σ_0\subseteq\text{Bad}_A$ with particular properties, namely $Σ_0$ is the set of places that do not divide the order of $A$ and ramify in the minimal extension splitting $A$.
